Embodiments of the present invention relate to interpolation and/or extrapolation of seismic data, for example marine seismic data.
As generally used in the art, and as used herein, interpolation relates to constructing one or more new data points within the range of a discrete set of known data points, whereas extrapolation relates to constructing one or more new data points outside the range of a discrete set of known data points.
Seismic exploration is widely used to survey the earth's interior, for example to locate and/or characterise hydrocarbon deposits. In simple terms, a seismic survey is carried out by causing one or more sources to emit seismic energy into the earth's interior. The seismic energy propagates downwardly into the earth, and is reflected by geological formations within the earth. The reflected seismic energy is detected by one or more detectors (generally known as “receivers”), and information about the structure of the earth's interior can be obtained from the seismic energy reflected to the receivers.
A seismic survey may be conducted on land, in water, or in a land-water transition zone. A seismic survey conducted in water is normally known as a “marine seismic survey”, although it should be understood that this term is not limited to a seismic survey carried out at sea (in salt water) but also covers surveys performed in fresh or brackish water.
FIG. 1(a) is a plan view and FIG. 1(b) is a side view of one form of a typical marine seismic survey, known as a towed marine seismic survey. Two alternately firing seismic sources 13, each comprising one or more seismic source elements, such as airguns, are towed by a survey vessel 11. Seismic receivers 101 are mounted on one or more receiver cables 10 (four cables are shown in FIG. 1(a), as an example), which are also towed through the water. FIGS. 1(a) and 1(b) show the receiver cables 10 as towed by the same survey vessel 11 as the sources 13, but in principle a second survey vessel could be used to tow the receiver cables 10. The receiver cables are often known as “seismic streamers”. A streamer may have a length of up to 12 km or greater, with receivers 101 being disposed every few metres along a streamer. A typical lateral separation (or “cross-line” separation) between two streamers in a typical towed marine seismic survey is of the order of 100 m but can often range from 25 m to 120 m. The area within which receivers are provided is known as the “acquisition aperture” of the survey.
Typically streamers are provided with one or more positioning systems for providing information about both the absolute and relative positions, of the streamers 10. For example, the streamers may be provided with sonic transceivers 112 for transmitting and receiving sonic or acoustic signals for monitoring the relative positions of streamers and sections of streamers. The streamers may alternatively or additionally be provided with a satellite-based positioning system, such as GPS, for monitoring the positions of the streamers—for example, GPS receivers (not shown) may be placed at the front and rear of each of the streamers. Similarly, GPS receivers and other positioning sensors are used to locate the source 13 relative to the streamers and in a suitable geodetic coordinate frame. Vertical and/or lateral adjustments to the streamer positions may be effected by steerable wings, or “birds” (111) located at intervals along the streamer and integrated with the positioning system through a suitable controller.
When a source, 13, is actuated, it emits seismic energy into the water, and this propagates downwards into the earth's interior until it undergoes reflection by some geological feature within the earth. The reflected seismic energy is detected by one or more of the receivers 101.
In a marine seismic survey, the receivers 101 typically comprise hydrophones and/or geophones. A hydrophone measures pressure (a scalar quantity), whereas a geophone measures particle velocity or acceleration (a vector quantity). A one-component (1-C) geophone measures particle velocity or acceleration along one direction, whereas a three-component (3-C) geophone measures particle velocity or acceleration along each of three mutually orthogonal directions. Today, modern marine seismic survey streamers may contain both hydrophones and 1-C, 2-C or 3-C geophones. Other geophone geometries may also be considered which span the three Cartesian dimensions with a minimum of three sensors and where four or more sensors provide redundancy and quality-control information.
Another type of marine seismic survey is a seabed seismic survey, in which the receiver cables, comprising both hydrophones and multi-component geophones, are disposed on the seabed rather than towed through the water.
In the survey of FIG. 1, the seismic source is typically actuated at regular distance intervals, and each actuation is known as a “shot”. When the source is actuated, the pressure is measured at each location where a hydrophone is provided on a streamer, and the particle velocity or acceleration (or a component thereof) is measured at each location where a geophone is provided on a streamer. This is repeated for each shot. It can therefore be seen that, for each shot, the pressure and/or particle velocity or acceleration is measured only at certain discrete locations within the acquisition aperture, corresponding to the positions of the hydrophones and geophones. Moreover, owing to the movement of the streamers through the water (which may be intentional owing to movement of the towing vessel 1 or the steerable birds 111, or may be unintentional owing to the action of wind, tides or currents), the positions at which the pressure and/or particle velocity or acceleration is measured for one shot will be different from the positions at which the pressure and/or particle velocity or acceleration is measured for another shot.
When acquired seismic data are processed it can often be desirable to know the pressure and/or particle velocity or acceleration at locations that are different from the positions at which the pressure and/or particle velocity or acceleration were measured. These locations may be inside the acquisition aperture of the receiver array, or they may be outside it. This is done by interpolating or extrapolating using the values of pressure or particle velocity/acceleration measured at the receiver positions to obtain estimates of the pressure or particle velocity/acceleration at locations different from the receiver positions. As one example, in a “time-lapse” seismic survey a survey is carried out at the same survey area at different times, for example to monitor a reservoir under production. Ideally each subsequent survey would be carried out with sources and receivers at the same positions as the earlier surveys, but this is very difficult to achieve with a towed marine survey. Notwithstanding the use of equipment, such as steerable birds 111, which allow a certain amount of control in the streamer positions to correct for cross-line drift of the streamers, surface currents can sometimes be too strong to allow a full correction to be made during data acquisition, so that the receiver positions in a subsequent survey do not exactly replicate the receiver positions in the initial survey. It is therefore usual to interpolate the pressure and/or particle velocity data recorded from each time-lapse survey onto a common acquisition grid, to allow the sets of data to be processed to give a time-lapse difference signal representative of changes in the reservoir.
The need to interpolate or extrapolate acquired seismic data is not limited to time-lapse seismic surveying, and there are many cases in which it is desired to interpolate or extrapolate acquired seismic data into regions where no receivers were present such as, for example, between adjacent receiver cables in the survey of FIG. 1. As a further example it may be desired to extrapolate the data to positions outside the acquisition aperture for the purpose of estimating data at very short offsets from the seismic source, which has application for attenuating multiples in the acquired data.
Various techniques for interpolating and extrapolating seismic data have been proposed. As an example, UK patent publication GB 2 414 299 discloses a technique for using measurements of pressure and gradients of pressure, or their equivalents such as components of particle velocity or acceleration, to interpolate the pressure field onto locations where no pressure measurements were actually made. The method of GB 2 414 299 uses a modified Taylor expansion to extrapolate the pressure field away from the two (or more) points and achieves the result through linear, Hermite or barycentric weighting, according to the application. The Taylor expansion is modified for interpolation according to a proposal from Kraaijpoel (Kraaijpoel, D., Seismic ray fields and ray field maps: theory and algorithms. PhD thesis, Utrecht University 2003) who defines weights which achieve an accuracy equivalent to one order higher than the order of the Taylor expansion.